Observers for Canonic Models of Neural Oscillators

D. Fairhurst¹, I. Tyukin^1,3,4*, H. Nijmeijer² and C. van Leeuwen³

¹ Department of Mathematics, University of Leicester, University Road, LE1 7RH, UK
² Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513 , 5600 MB, Eindhoven, The Netherlands
³ RIKEN (Institute for Physical and Chemical Research) Brain Science Institute, 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan
⁴ Deptartment of Automation and Control Processes, St-Petersburg State University of Electrical Engineering, Prof. Popova str. 5, 197376, Russia

^* Corresponding author. E-mail: I.Tyukin@le.ac.uk

Abstract

We consider the problem of state and parameter estimation for a class of nonlinear oscillators defined as a system of coupled nonlinear ordinary differential equations. Observable variables are limited to a few components of state vector and an input signal. This class of systems describes a set of canonic models governing the dynamics of evoked potential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo, and Morris-Lecar models. We consider the problem of state and parameter reconstruction for these models within the classical framework of observer design. This framework offers computationally-efficient solutions to the problem of state and parameter reconstruction of a system of nonlinear differential equations, provided that these equations are in the so-called adaptive observer canonic form. We show that despite typical neural oscillators being locally observable they are not in the adaptive canonic observer form. Furthermore, we show that no parameter-independent diffeomorphism exists such that the original equations of these models can be transformed into the adaptive canonic observer form. We demonstrate, however, that for the class of Hindmarsh-Rose and FitzHugh-Nagumo models, parameter-dependent coordinate transformations can be used to render these systems into the adaptive observer canonical form. This allows reconstruction, at least partially and up to a (bi)linear transformation, of unknown state and parameter values with exponential rate of convergence. In order to avoid the problem of only partial reconstruction and at the same time to be able to deal with more general nonlinear models in which the unknown parameters enter the system nonlinearly, we present a new method for state and parameter reconstruction for these systems. The method combines advantages of standard Lyapunov-based design with more flexible design and analysis techniques based on the notions of positive invariance and small-gain theorems. We show that this flexibility allows to overcome ill-conditioning and non-uniqueness issues arising in this problem. Effectiveness of our method is illustrated with simple numerical examples.